Finite Group Databasebeta

← #99623

Subgroups — #99623

Order 512.

510 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23755C211yesmaximal, characteristic
2#22170C211yesmaximal, characteristic
3#11370C211yesmaximal, characteristic, complemented
4#8710C211yesmaximal, characteristic, complemented
5C4×#327C211yesmaximal, characteristic
6#7028C211yesmaximal, characteristic, complemented
7#7026C211yesmaximal, characteristic, complemented
8C2×#435—21no—
9C4×#121C2211yescharacteristic
10#2285C2211yescharacteristic
11#2100C411yescharacteristic, complemented
12#2094C411yescharacteristic, complemented
13#2084C411yescharacteristic, complemented
14#2077—21no—
15#2071C411yescharacteristic, complemented
16#2034—21no—
17#1924—21no—
18#1917—21no—
19#1870—21no—
20#1869—21no—
21#1850—21no—
22#1797C2211yescharacteristic
23C4×#106—21no—
24C4×#102—21no—
25C4×C22⋊C8C2211yescharacteristic
26C4×C22⋊C8C2211yescharacteristic
27C2×#339—21no—
28C2×#339—21no—
29#1728C2211yescharacteristic
30C2×#327—21no—
31C4×#95C2211yescharacteristic
32#1294—21no—
33#1279—21no—
34#1279—21no—
35C22×M16D411yescharacteristic
36C22×M16D411yescharacteristic
37C23×C8—21no—
38C2×#121D411yescharacteristic
39C2×C4×D4—21no—
40C2×C4×D4D411yescharacteristic
41C2×#117—21no—
42C22×C42—21no—
43C22×C42C2311yescharacteristic
44#435—42no—
45#435—42no—
46C2×#106—21no—
47C2×#106—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8D411yescharacteristic
50C2×C4⋊C8D411yescharacteristic
51C2×#102—21no—
52C2×#102—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—21no—
59C2×#97—21no—
60C2×C4×C8—21no—
61C2×C4×C8—21no—
62#390C2×C411yescharacteristic
63#389C2×C411yescharacteristic
64#388—21no—
65#386—21no—
66#384C2×C411yescharacteristic
67#384C2×C411yescharacteristic
68#382—21no—
69#380—21no—
70#380—21no—
71#380D411yescharacteristic
72C4×C4⋊C4—21no—
73C4×C4⋊C4—21no—
74C4×C4⋊C4D411yescharacteristic
75C4×C22⋊C4—21no—
76C2×#95—21no—
77C2×#95—21no—
78#340—21no—
79#340—21no—
80#340—21no—
81#340—21no—
82#339—42no—
83#339—42no—
84#336—21no—
85#336—21no—
86#336—21no—
87#336—21no—
88#336—21no—
89#336—21no—
90#336C2×C411yescharacteristic
91#336C2×C411yescharacteristic
92#327—24no—
93C22×D4—21no—
94C22×D4C22⋊C411yescharacteristic
95C23×C4—21no—
96C23×C4—41no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—21no—
101C2×M16—21no—
102C2×M16—21no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—41no—
107C2×M16SD1612yes—
108C22×C8—21no—
109C22×C8—21no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—42no—
115C22×C8—41no—
116C22×C8—81no—
117C22×C8—81no—
118#121—22no—
119#121—41no—
120C4×D4—41no—
121C4×D4—41no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126#117—41no—
127#117—41no—
128#117—41no—
129#117—41no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4C22⋊C411yescharacteristic
138C2×C4⋊C4C22⋊C411yescharacteristic
139C2×C4⋊C4C22⋊C411yescharacteristic
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—41no—
146C2×C42—21no—
147C2×C42—22no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42C22×C411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42#4111yescharacteristic
160#106—42no—
161#106—42no—
162C4⋊C8—41no—
163C4⋊C8—41no—
164C4⋊C8—42no—
165#102—42no—
166#102—42no—
167C22⋊C8—22no—
168C22⋊C8—42no—
169C22⋊C8—42no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—81no—
174C22⋊C8—81no—
175#97—42no—
176C4×C8—41no—
177C4×C8—41no—
178C4×C8—41no—
179C4×C8—41no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—22no—
184#95—22no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189C24—21no—
190C24#12111yescharacteristic
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—42no—
195C2×D4—41no—
196C2×D4—41no—
197C2×D4—81no—
198C2×D4#10412yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4#12011yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12311yescharacteristic
235C22×C4C2×SD1611yescharacteristic
236C22×C4#13511yescharacteristic
237C22×C4AGL(1,ℤ/8)11yescharacteristic
238C22×C4#13711yescharacteristic
239M16—42no—
240M16—42no—
241M16—81no—
242M16—81no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—84no—
263C2×C8—81no—
264C2×C8—81no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C22⋊C4—81no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—41no—
297C42—41no—
298C42—42no—
299C42—42no—
300C42—41no—
301C42—41no—
302C42—81no—
303C42—81no—
304C42—81no—
305C42—81no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—22no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23#38611yescharacteristic
320C23#46111yescharacteristic
321C23#46311yescharacteristic
322D4—42no—
323D4—81no—
324D4—82no—
325D4—82no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—44no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4C2×#10411yescharacteristic
377C2×C4#42111yescharacteristic
378C2×C4C4×SD1611yescharacteristic
379C2×C4#44011yescharacteristic
380C2×C4#44311yescharacteristic
381C2×C4#44411yescharacteristic
382C2×C4#45011yescharacteristic
383C2×C4#45211yescharacteristic
384C2×C4#45311yescharacteristic
385C2×C4#45411yescharacteristic
386C2×C4#45811yescharacteristic
387C2×C4#45811yescharacteristic
388C8—42no—
389C8—42no—
390C8—81no—
391C8—82no—
392C8—81no—
393C8—81no—
394C8—81no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22#187111yescharacteristic
413C22#187611yescharacteristic
414C22#187711yescharacteristic
415C22#188111yescharacteristic
416C22#188311yescharacteristic
417C22#188411yescharacteristic
418C22#189011yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—41no—
426C4—41no—
427C4—84no—
428C4—81no—
429C4—81no—
430C4—81no—
431C4—161no—
432C4—161no—
433C4#161311yescharacteristic
434C4#161511yescharacteristic
435C4#161611yescharacteristic
436C4#161811yescharacteristic
437C2—41no—
438C2—41no—
439C2—82no—
440C2#896911yescharacteristic
441C2#898911yescharacteristic
442C2#899711yescharacteristic
443C2#901111yescharacteristic
444C2#903811yescharacteristic
445C2#904511yescharacteristic
446C2#905511yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #11370C2semidirect
4. #8710C2semidirect
6. #7028C2semidirect
7. #7026C2semidirect
11. #2100C4semidirect
12. #2094C4semidirect
13. #2084C4semidirect
15. #2071C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#237552
#221702
#113702
#87102
C4×#3272
#70282
#70262